Public Signals, Voluntary Disclosures, andSecurity Prices

Davide Cianciaruso, Dor Lee-Lo and Sri S. Sridhar1

Work-in-progress & incomplete

March 22, 2018

1Department of Accounting and Management Control, HEC Paris, School of Management, TelAviv University, and Kellogg School of Management, Northwestern University, respectively.

Abstract

This paper establishes that when a �rm borrows, its manager interested in maximizing equity

price is likely to disclose his private information more often if and only if the posterior default

probability decreases following a public signal.

When both posterior mean and variance vary in the realized value of the public signal, we

�nd that the posterior default probability is non-monotonic in public news. As a consequence,

each of posterior probability of the manager�s discretionary disclosure and the �rm�s equity

price can be decreasing in the realized value of the positively correlated public news.

Applications of this trade-o¤ between the posterior variance e¤ect versus the mean e¤ect

generate several signi�cant predictions about the impact of mandatory disclosures and ana-

lysts�forecast consensus and dispersion on �rms�discretionary disclosures and equity prices.

In particular, we predict that a mandatory disclosure requirement would actually decrease

the voluntary disclosure likelihood for su¢ ciently large and small values of public news if

prior default probability is less than 0:5.

Finally, most of above results apply qualitatively to an all-equity �rmwith limited liability

for equityholders.

1 Introduction

Publicly listed �rms often experience occurrence of a variety of signi�cant public signals

about them. Examples of such public signals with major economic signi�cance would include

�rms�own mandatory disclosures such as earnings reports, analysts� forecasts, regulators�

actions and statements such as the Food and Drug Administration�s (dis) approval of a

drug application and SEC�s investigations, news of a large proposed merger/acquisition and

subsequent regulator�s approval or denial of such proposal (e.g., recent Broadcom�s bid for

Qualcom, etc.,) public news of data leaks caused by hacking (e.g., Equifax), social media

responses to a �rm�s proposed major plan, consumers� responses to �rms� products and

services such as the product market�s reception to a new model of Apple�s iphone, product

recalls or hazards, development of a new product, process or technology (e..g., Boeing�s 787

or Tesla�s new model announcement), major law suits or judgements related to signi�cant

law suits, a highly visible or reputed manager�s departure from or joining a given �rm, etc.

Acharya, DeMarzo, and Kremer�s, 2011, (ADK) theoretical analysis �nds that any such

public signals would not change �rms�voluntary or discretionary disclosure probabilities,

regardless of whether such public signals are favorable or unfavorable and irrespective of

the magnitude of their economic signi�cance to the �rms. ADK�s result arises from capital

markets recalibrating their prior beliefs about distributions of �rms� cash �ows based on

such public signals in a manner that in equilibrium �rms�voluntary disclosure probabilities

would not change.

However, there is a fair amount of empirical literature that does demonstrate �rms�vol-

untary disclosure strategies being in�uenced by a variety of public signals for a wide set

of reasons. For instance, Lansford [2006] �nds �rms tend to make voluntary disclosures of

favorable patent information more often following negative earnings surprises.1 This empiri-

cally documented phenomenon of voluntary disclosure behavior of �rms being in�uenced by

prior or concurrent public signals �nds limited theoretical explanation.

ADK�s [2011] result of the indi¤erence of �rms�voluntary disclosure proclivities to prior

public news is obtained in a setting with unlimited liability for equityholders. A signi�cant set

of prior voluntary disclosure assumes similar unlimited liability for equityholders. However,

it is a well-known legal fact that shareholders of incorporated �rms do enjoy limited liability.

So, do debtholders of all �rms. This paper, therefore, examines whether such prior voluntary

1Other papers that �nd similar evidence of �rms� voluntary disclosure strategies being in�uenced bypublic news for several other reasons include �rms reducing their disclosures in response to litigation events(Rogers and Van Buskirk, 2009), changes in market characteristics such as decreasing market uncertaintyfollowing earnings announcements mitigated by voluntary forecasts possibly leading to greater stock pricevolatility (Rogers, Skinner and Van Buskirk, 2009), and voluntary management forecasts being in�uencedby earnings reports (Rogers and Van Buskirk 2013.)

1

disclosure results can be extended to settings with limited liabilities.2 However, we know

that the introduction of limited liability induces call-option-like features in equity price

behavior.3 It, therefore, further appears natural that any examination of �rms�voluntary

disclosure behavior in limited liability settings should also allow for markets to update both

unknown mean and unknown variance to better understand �rms� optimal discretionary

disclosure responses to such public signals and consequent securities price behavior.

We examine a setting in which a (possibly levered) �rm�s cash �ow is stochastically

distributed with unknown mean and unknown variance. The markets update their posterior

beliefs about both the mean and the variance based on a public signal which is correlated

with the �rm�s cash �ow. The manager observes private information about the �rm�s cash

�ows with some positive probability. One major objective is to understand how in the

presence of limited liability, the manager�s discretionary disclosure decision responds to the

public signal and how security prices vary in the public signal.

For a broad class of distributions, our analysis predicts that the public signal increases

the levered �rm�s discretionary disclosure probability if and only if the posterior default

probability following the public signal decreases. This prediction is somewhat surprising for

several reasons. First, the manager�s objective in deciding on whether to make a discre-

tionary disclosure is only to maximize the �rm�s expected equity price. Yet, his disclosure

probability increases in the realized value of the public signal if and only if the posterior

default probability decreases. As will be seen below, the posterior default probability is not

monotone in the public signal. So, when the public signal is positively correlated with the

�rm�s cash �ows, one might presume that greater public news should increase the manager�s

likelihood of voluntary disclosure. Our result negates this intuition. Second, this result

contrasts with ADK. [2011], who demonstrated that a public signal preceding a voluntary

disclosure decision should not a¤ect the voluntary disclosure probability because the original

priors are recalibrated following the public signal in a manner that, with unlimited liability

for equityholders, the manager�s voluntary disclosure probability would not change. In our

setting, this inverse relation between the posterior default probability and the manager�s

discretionary disclosure probability as a function of the realized value of the public signal

2Sridhar and Magee [1996] examine earnings reports in the context of limited liability. Fischer andVerrecchia [1997] examine the impact of earnings reports on the behavior of equity and debt prices in thepresence of such limited liability.

3There is a large amount of empirical research which have examined non-linear price behavior and theimpact of debt default probability on equity price. Papers that examine such price behavior include Freemanand Tse [1992], Subramanyan and Wild [1993], Das and Lev [1994], Dhaliwal and Reynolds [1994], Hayn[1995], Bao and Datta [2014], Bischol et al. [2016], Campbell et al. [2014], Hope et al. [2016], etc.However, most prior theoretical voluntary disclosure literature (including the one that examines such

behavior in the presence of limited liability) focuses on updating only mean for a given known variance.

2

arises only because of the limited liability. Third, this result is independent of whether the

posterior precision of the cash �ow distribution is invariant with respect to the realized value

of the public signal (e.g., as in the case of a normal distribution).

Another prediction that arises immediately from this inverse relation between the pos-

terior default probability and the voluntary disclosure probability is that the probability

of voluntary disclosure decreases in the amount of leverage. If the �rm were not levered,

this necessary and su¢ cient condition would still hold if the equity holders enjoy limited

liability. The requirement that the equity price must be greater than or equal to zero in

the presence of limited liability of equityholders would also yield the result that the �rm�s

voluntary disclosure probability varies in the realized value of the public signal.

We next proceed to examine a setting which allows for the market�s posterior beliefs

about both the mean and the variance to change in the realized value of the public signal.

We assume that the cash �ow realized at the end of the game and the interim public signal are

jointly distributed according to a multivariate student�s t distribution. De�ning the public

"news" as the magnitude of the surprise in the public signal, we �nd that with a normal

distribution assumption for cash �ows, the posterior equity value following the public signal

is monotonically negatively related to the posterior default probability. We next proceed

to examine a setting with a student�s t distribution that allows for the posterior variance

also to change in the realized value of the public news. Our analysis predicts that the

posterior equity price and default probabilities are positively correlated following su¢ ciently

good and bad public news, provided the prior default probability is less than 0:5: That the

equity price can actually increase in the posterior default probability may at �rst blush seem

counterintuitive, but is best understood from examining the interaction between updating

the mean versus variance based on the public signal. With fatter tails, a large surprise (either

good or bad) in public news would increase the posterior variance in a signi�cant manner.

Consequently, when prior beliefs about the default probability are lower, then the posterior

variance e¤ect dominates the posterior mean e¤ect for su¢ ciently good or bad public news,

leading to the equity prices being positively correlated with posterior default probability.

However, where the prior default probability is greater than 0:5; even moderately bad

public news can yield similar positive relation between the posterior default probability

and equity prices. The intuition for this seemingly surprising result is that while the prior

mean e¤ect serves to increase the posterior default probability for moderately bad public

news, the relatively low posterior variance is su¢ cient to increase the �option value�e¤ect

embedded in equity prices on account of limited liability. In this way, prior beliefs about

default probability continue to exert signi�cant in�uence on the nature of relation between

the posterior default probability and equity prices as a function of the the realized value of

3

the public news.

These results concerning positive relation between the posterior default probability and

the equity price vanish when the posterior variance does not change in the realized value

of the pubic signal, as in the case of a normal distribution. This comparison highlights

the important role played by the markets updating the variance in establishing the relation

between (posterior) default probabilities and equity prices.

Our analysis further predicts that the posterior probability of a voluntary disclosure,

following a public signal, always increases in the prior mean of the cash �ow. However,

whether the posterior probability of voluntary disclosure following the public signal would

increase in the prior volatility of the cash �ow depends on the magnitude of prior default

probability. The posterior probability of voluntary disclosure actually decreases in the prior

cash �ow volatility if and only if the prior default probability is less than 0:5:We also predict

that the posterior probability of voluntary disclosure decreases (increases) in the correlation

between the public signal and the cash �ow for relatively low (high) realized values of the

public signal.

A major �nding of our analysis is that the posterior default probability is non-monotonic

in the public signal. When the prior default probability is less than 0:5; the posterior

default probability is U�shaped; then, there exists a threshold level of pubic signal at whichthis posterior default probability reaches its minimum level, so that the posterior default

probability is increasing and voluntary disclosure is decreasing for public signal values greater

than this threshold. In contrast, when the prior default probability is more than 0:5; the

posterior default probability is hump�shaped; then, there exists another threshold level ofpubic signal at which this posterior default probability reaches its maximum level, so that the

posterior default probability is increasing and voluntary disclosure is decreasing for public

signal values less than this second threshold level. In this way, the prior default probability

determines the shape of the posterior default probability following public signal, which shape

in turn in�uences the posterior voluntary disclosure probability.

Moreover, our analysis predicts that either when the stochastic cash �ows and public sig-

nals are jointly normally distributed or in the knife-edge case of the prior default probability

being exactly equal to 0:5, the posterior default probability is monotonically decreasing and

posterior voluntary disclosure probability is monotonically increasing in the realized value of

the public signal.

We apply the conceptual �ndings from our general analysis to two major areas of interest:

interpreting public signals as mandatory disclosures and as analysts�forecasts. Our �rst set

of results concerns with how the mandatory disclosures of a noisy signal of the �rm�s cash �ow

a¤ect the equity price behavior, particularly when not followed by (or a concurrent) voluntary

4

disclosure. Our analysis predicts that the price for any given realized value of mandatory

disclosure, followed by no voluntary disclosure, can increase in the prior mean of the cash

�ow and over regions of relatively less favorable values mandatory disclosures particularly

when posterior default probability decreases. Such non-disclosure prices can also increase

in the prior cash �ow volatility when prior default probability is weakly greater than 0:5;

but can decrease in the correlation between the cash �ow and the public signal, particularly

when the mandatory disclosure is relatively unfavorable. Further, our analysis predicts that

the introduction of a mandatory disclosure requirement can actually decrease voluntary

disclosure probabilities for su¢ ciently good and bad values of mandatory disclosures when

prior default probability is less than 0:5: These results are driven by the markets updating

both the unknown mean and variance based on the mandatory disclosures. These Einhorn

[2005] also obtains similar predictions about how mandatory disclosures can reduce voluntary

disclosures in the presence of multiple private correlated signals, one of which is required to

be disclosed mandatorily.

Finally, when we view analysts�forecast reports as a set of public signals, our analysis

�rst predicts that the posterior mean of the �rm�s cash �ow conditioned on the entire set of

analysts�forecast reports can be reduced to a function of the analysts�forecasts consensus.

In other words, the analysts�forecast consensus (as opposed to the entire vector of analysts�

reports) is su¢ cient to derive the posterior mean of the �rm�s cash �ows. Similarly, the

analysts�forecast consensus and dispersion together are su¢ cient (again, as opposed to the

entire set of analysts�reports) to determine the bayesian-updated posterior variance of the

�rm�s cash �ows. Our analysis further predicts that the posterior default and posterior

voluntary disclosure probabilities are non-monotonic in the analysts�forecasts consensus; as

a consequence, the posterior default probability actually increases and posterior voluntary

probability actually increases in the consensus when prior default probability is greater than

0:5 and for su¢ ciently high values of consensus. Moreover, the �rm�s posterior default

probability is predicted to actually decrease (increase) in forecast dispersion if prior default

probability is greater (less) than 0:5. Finally, we also predict that the �rm�s posterior

voluntary probability is predicted to actually increase (decrease) in forecast dispersion if

prior default probability is greater (less) than 0:5.

While Sridhar and Magee [1996] examine manager�s opportunistic disclosure management

in the context of a debt contract, they do not examine the role of a public signal in the

presence of limited liability. Fischer and Verrecchia [1997] examine the impact of public

signals in the presence of limited liability. In contrast, we examine how the presence of

public signals a¤ect the �rm�s voluntary disclosure probabilities and generate a variety of

predictions related to multiple applications of our setting. Prior literature that examines the

5

updating on variances include Subramanyam [1996], Jorgensen and Kirschenheiter [2003],

Beyer [2009] and Heinle and Smith [2017]. Subramanyam [1996] was perhaps the �rst

paper to show how posterior precision behaves for a fairly general class of distributions, and

consequently, how stock returns exhibit non-linear properties in the surprise component of a

public signal. While Jorgensen and Kirschenheiter [2003] examine the voluntary disclosure

behavior of variances when such disclosure is costly, Beyer [2009] examines a setting in

which the manager optimally manages both the forecast and earnings reports when the

markets update both unknown mean and variance Heinle and Smith [2017] examine how

a �rm�s imperfect public signal about the variance of the cash �ow would lead to variance

uncertainty premium when the market knows the mean, but updates unknown variance.

They proceed to demonstrate that the �rm�s cost of capital decreases because the �rm�s

public signal reduces the variance uncertainty premium. In contrast to these papers, we

examine how a manager�s voluntary disclosure strategy and price behavior are a¤ected by a

public signal when the market updates both unknown mean and variance.

2 The Model

Our goal is to study how public information a¤ects the voluntary disclosures of �rms in

the presence of limited liability in general and, more speci�cally, for levered �rms. Assume

that at time t = 0; an incorporated �rm must borrow $M to invest in a project which

generates a random cash �ow X at time t = 3:4 Normalizing the expected rate of return for

the debtholders to zero, the face value � > 0 is determined as the solution to the following

participation constraint for lenders:

E [Vd (X)] =M;

where the value of the debt given X is determined as

Vd (X) � max fmin fX; �g ; 0g

Similarly, the equity value as a function of X is given by

VE (X) � max fX � �; 0g :

The expressions for equity and debt values above re�ect the legal fact that owners of and

4Throughout, we use capital letters to denote a random variable (e.g., X) and small letters to denotetheir realization (e.g., x).

6

lenders to incorporated �rms enjoy limited liability. It follows that time equity and debt

prices at the beginning of the game (i.e., at time t = 0) are given by

P0E (�) = E [max fX � �; 0g] ; andP0E (�) = E [max fmin fX; �g ; 0g] ;

respectively.

At time t = 1; a public signal Y about X is realized. At a later point in time, say, at time

t = 2; we assume that (following Dye,1985) with probability q 2 (0; 1) the manager privatelyobserves a perfect signal about the �rm�s liquidating dividend, X. If informed, the manager

can publicly disclose his private information. We assume that any such voluntary disclosure

must be truthful. Also, we assume that an uninformed manager cannot credibly convey the

fact that he received no private information about X. The goal of the manager is to choose

disclosure of the realized X = x (denoted (d; x)) or non-disclosure (denoted nd) in order to

maximize the market price of equity at time t = 2; P2E (:). The public information set at

time t = 2 is I = fy; xg, if the manager discloses, and I = fy; ndg, if the manager does notdisclose.

Figure 1 below provides the time-line for our setting.

t = 0 t = 1 t = 2 t = 3

The �rm borrows A public signal The �rm�s manager observes The �rm realizes

$M by entering Y is realized. a private signal X with prob. its liquidating

into a debt contract Equity price q 2 (0; 1) and decides on a dividend X:

with face value of P1E (Y ) and discretionary disclosure.

debt �: debt price Equity price P2E (I) andP1D (Y ) form. debt price P2D (I) form.

In a market populated by risk-neutral investors, the price of equity at time t = 2; P2E (I)given the public information set I is

P2E (I) = E [VE (X) jI] ;

that is the posterior expected value of equity conditioned on all publicly available information

at time t = 2.

We make the following assumption on the joint distribution of (X;Y ).

7

Assumption 1 The distribution of X conditional on the realized Y = y can be represented

as follows:

Xj (Y = y) = � (y) + � (y)U; (1)

where � (�) and � (�) are deterministic functions and the random variable U is zero-mean,

continuously distributed with a positive density over the real line, and independent of Y .

That is, we assume that the realized public signal a¤ects potentially both the posterior

mean and variance of the liquidating value. For the moment, we do not impose any re-

strictions on � (�) and � (�) other than � (y) > 0 for all y. This condition guarantees that

observing Y never resolves all the uncertainty about X. The representation of the posterior

distribution in (1) is rather general, and can be recovered from various (commonly used)

joint distributions of (X; Y ). Two important examples are the multivariate normal and mul-

tivariate Student�s t distribution. Our �rst major result, Proposition 1, relies only on the

broad distributional structure in assumption (1).

3 Default probability and voluntary disclosures

The objective of this section is to understand how the limited liability structure a¤ects the

manager�s equilibrium discretionary disclosure behavior. Fix a public signal Y = y. It is

straightforward to establish that there exists a disclosure threshold x̂ (y) such that the set of

realizations of X that are voluntarily disclosed in equilibrium must be an interval of the form

[x̂ (y) ;1) : This (usual) result follows from the disclosure payo¤, P2E (y; x), weakly increasingin the realization X = x while the non-disclosure payo¤, P2E (fy; ndg), is independent ofx. Further, note that the disclosure threshold x̂ must be such that x̂ > �. Otherwise, the

disclosure payo¤ for x < � would be zero even though the non-disclosure payo¤ is strictly

positive. This latter result follows from a strictly positive posterior weight to the possibility

of the manager being uninformed and that posterior expected residual claims of equityholders

conditioned on the manager not being informed are strictly positive, i.e., Pr(the manager

not being informedjnd)�E�(X � �)+ jY = y; the manager not informed

�> 0.

Therefore, the disclosure threshold x̂ must satisfy the indi¤erence condition:

x̂ (y)�� =(1� q)E

�(X � �)+ jY = y

�+ q Pr [X < x̂ (y) jY = y]E

�(X � �)+ jY = y; x < x̂ (y)

�(1� q) + q Pr [X < x̂ (y) jY = y] ;

(2)

where fXjY=y (x) denotes the density function of XjY = y which is used to determine the

above posterior expectations. The left hand side of the above equation represents the payo¤

from disclosing X = x̂ (y) and the right hand side denotes the date-2 price given the public

8

signal Y = y and the �rm�s non-disclosure, P2E (fy; ndg) :For any given face value of debt �; de�ne the posterior default probability following the

public signal Y = y as Pr (X < �jy; �) :When equity holders have unlimited liability, Acharya, Demarzo and Kremer [2011] show

that any public signal merely recalibrates the prior beliefs, and hence, the manager�s dis-

closure likelihood does not vary in the realized value of the public signal Y: In contrast,

Proposition 1 below predicts that the manager�s voluntary disclosure likelihood is inversely

related to the posterior default probability, where such posterior default probability is con-

ditioned on the public signal Y: In other words, even though the manager�s objective is to

maximize the �rm�s equity price at time t = 2; a necessary and su¢ cient condition for the

manager�s disclosure probability to increase is a decreased posterior default probability. All

proofs are in the Appendix.

Proposition 1 The realized public signal Y = y increases the posterior probability of vol-

untary disclosure of the manager�s private information X if and only if it decreases the

posterior default probability, Pr [X < �jY = y].

Here it must be noted that the �rm need not be a levered �rm to obtain the above result.

In other words, Proposition 1 result would hold even for an all-equity �rm, where the legal

construct of limited liability for equityholders of incorporated �rms would mean � � 0: Forany given public signal Y = y; de�ne the standardized disclosure threshold and face value

by

~x (y) � x̂� � (y)� (y)

; ~� (y) � � � � (y)� (y)

The intuition for Proposition 1 result follows from noting that the derivatives of ~x (y) and~� (y) in the realized value of the public signal y have the same sign regardless of how the

functions � (y) and � (y) vary in y subject to our assumption that � (y) > 0 for all y: Using

the distributional structure in (1) in the indi¤erence condition (2) ; the result in Proposition

1 follows.

One might have imagined that given the manager�s objective function of maximizing the

equity price P2E (�) ; the posterior disclosure probability would also be directly in�uenced byposterior expected residual claims of equity holders after paying the debtholders�claims for

any given public signal Y = y: However, Proposition 1 establishes the somewhat surprising

result that the manager�s discretionary disclosure behavior may seem to mimic more the

debtholders�interests given the result that increasing disclosure likelihood as the posterior

default probability decreases.

9

Acharya et al., (2011) point out that the release of a public signal prior to the manager�s

discretionary disclosure decision has a twofold e¤ect on the probability of information with-

holding: a distribution e¤ect and a threshold e¤ect. The former e¤ect consists in the fact

that, for a �xed threshold x̂, the realized y changes the posterior mean � (y) and volatility

� (y) of X, thereby changing the probability of the �rm withholding. The latter is an e¤ect

that occurs because a di¤erent posterior distribution of X implies a di¤erent equilibrium

threshold; in other words, ~x (y) itself is a function of the public news. In voluntary disclo-

sure models with unlimited liability such as Acharya et al. (2011), the two e¤ects exactly

o¤set each other. Here, on the contrary, the distribution e¤ect dominates the threshold

e¤ect. The disclosure decision is taken to maximize the equity value, max fX � �; 0g. Froma statistical point of view, the residual claim to equity holders is a censored version of the

random variable X. Therefore, equity prices respond less starkly to the public news than

does the distribution of X, leading to the result that a greater posterior default probability

discourages voluntary disclosures.

Acharya and Johnson [2007] �nd a greater incidence of insider trading in those credit

default swap (CDS) markets for those securities which are characterized by greater credit

risks. This empirical �nding is consistent with our prediction of equilibrium disclosure be-

havior. A higher posterior default probability would more likely lead to a greater credit risk.

Less frequent voluntary disclosures under such circumstances would facilitate more aggres-

sive insider trading by the manager to take advantage of the consequent greater information

asymmetry in the market. Further, Proposition 1 result provides one possible theoretical

explanation for the empirical �nding in Drucker and Puri [2009] of a negative association

between the likelihood of a loan sale by banks and the distance-to-default of the client �rm.

The bank that had lent to this �rm initially at t = 1 is more likely to obtain superior private

information from monitoring the client following a higher posterior default rate. Conse-

quently, the bank is more likely to sell its asset in the secondary market more often given the

greater likelihood of non-disclosure by the �rm. In this way, the �rm�s increasing reluctance

to make voluntary disclosures in the posterior default probability would encourage the bank

to sell its asset more frequently in the secondary market.

A direct consequence of the result in Proposition 1 generates the following prediction.

Remark 1 The posterior probability of voluntary disclosure is decreasing in the amount ofborrowing M for every realized value of the public news, Z = z:

As the amount of borrowing M increases, ceteris paribus, the face value of � would

increase. This in turn causes the posterior default probability to increase, thus leading to

lower frequency of voluntary disclosures.

10

An immediate implication of the above corollary is that following any given public signal,

an all-equity �rm is more likely to make a voluntary disclosure than any levered �rm.

4 Impact of public signal on default probability, secu-

rity prices and voluntary disclosures

Recollect that the result in Proposition 1 only relied on the distributional structure in as-

sumption (1) : To model the notion of an economically signi�cant public signal that would

allow the manager and the markets to update both the mean and the variance of the �rm�s

cash �ows, we make the following distributional assumption.5

Assumption 2 (X; Y ) are jointly distributed according to a bivariate t distribution (denotedt2) with k � 3 degrees of freedom,6

X

Y

!� t2

�X

�Y

!;

�X �XY

�XY �Y

!; k

!;

where �XY � �XYp�X�Y

is such that 0 < �XY < 1.

As is well known (DeGroot, 1970), the variance-covariance matrix of the joint t distrib-

ution (of the �rm�s cash �ows X and the public signal Y ) is given by

k

k � 2

�X �XY

�XY �Y

!:

Therefore, the inequality �XY > 0 requires that the liquidation value and the public signal

are positively correlated. This assumption allows us to interpret a higher realization of the

public signal as better news.7 The correlation between X and Y , denoted �XY , is assumed

to be less than one in absolute value, so that Y is not a perfect signal of X.

De�ne as �news�Z � ��12

Y (Y � �Y ) as the magnitude of the surprise in the public signal(Y � �Y ) normalized by its standard deviation,

p�Y .

5If the equityholders were to be subject to unlimited liability, then even with our student�s t distributionassumption we would get the linearity of equity price in public signal Y and the probability of voluntarydisclosure being independent of the public signal, results that are well known in other contexts, e.g., Acharya,DeMarzo and Kremer [2011[.

6We require at least three degrees of freedom to ensure that the variance-covariance matrix is well-de�ned.Further, note that in the limiting case, as the degrees of freedom k !1; the t distribution approaches a

normal distribution.7Assuming �XY > 0 is also without loss of generality, because if the correlation were negative we would

adopt the opposite convention that a lower realization of Y is better news.

11

Remark 2 The random variable XjY = y can be represented as in (1):

Xj (Z = z) = � (z) + � (z)Uk+1; (3)

where Uk+1 denotes a standard univariate t distribution with k + 1 degrees of freedom,

� (z) = �X + �XY �Xz; and

� (z) = �X

r(1� �2XY )

k + z2

k + 1:

In the above, �X �p�X parametrizes the standard deviation of the liquidation value (for

�xed degrees of freedom k).8

Using these de�nitions, the posterior expected equity value conditioned on the normalized

values of the public signal are

E [VE (X) jZ = z] = � (z)Z 1

~�

�u� ~�

�gk (u) du; (4)

where ~� � ���(z)�(z)

and ~� � �(z)�(z)

are �standardized�versions of the face value of the debt and of

the mean, respectively, and gk (�) is the density of the standard univariate t distribution forgiven k degrees of freedom. Expressions (4) and (??) are derived in the proof of Proposition2. Also, the probability of default conditional on public news after normalization can be

expressed as:

Pr [X < �jZ = z] = Gk�~��:

We next generate predictions of how date�1 equity prices relate to posterior defaultprobabilities as functions of both prior beliefs as at t = 0 and date-1 public signal Y:

Proposition 2 (a) When only the posterior mean (and not the posterior variance) of cash�ow X changes in the realized value of the public signal Y (i.e., when k !1):

The posterior equity value at t = 1; P1E (z) ; is negatively related to the posterior

default probability at t = 1:

(b) When both the posterior mean and posterior variance of cash �ow X change in the

realized value of the public signal Y (i.e., when k <1):(b.i) If the prior default probability is lower than 0:5, then the posterior eq-

uity price at t = 1; P1E (z) ; and the posterior default probability are positively related for

8The actual standard deviation isq

kk�2�X , because the marginal of X is a t distribution, so �X is the

standard deviation of X only in the limiting case of the normal distribution (k !1).

12

su¢ ciently good and bad public news z; and

(b.ii) If the prior default probability is greater than 0:5; the posterior equity value

at t = 1; P1E (z) ; and the posterior default probability are positively related for moderately

bad public news z.

The intuition for the above predictions follows. Recollect that when the degrees of free-

dom k !1; the student�s t-distribution approaches the normal distribution, in which casethe posterior variance conditioned on the realized value of public news Z is independent of

such Z: So, the posterior distribution shifts to the right in the sense of �rst order stochas-

tic dominance leading to increases in posterior expected equity value and a decrease in the

posterior default probability. Hence, we get the result in Proposition (2a) :

However, when k <1; the posterior mean and variance move in fairly complex ways inpublic news. As a consequence, the relationship between posterior equity prices and posterior

default probability are not as simple as in the case of a normal distribution updating. First,

observe from Remark (2) that the posterior variance increases in the magnitude of publicly

disclosed surprise, z. When prior beliefs are optimistic (i.e., when prior default probability

is lower than 0:5), the posterior mean e¤ect is dominated by posterior variance e¤ect both

for su¢ ciently large favorable and unfavorable public news. Hence, the posterior volatility

e¤ect can dominate the posterior mean e¤ect for su¢ ciently high values of public news z;

with the result that both the �option-type�characteristic of equity value and the posterior

probability of default can increase at the same time in z. This dominating posterior volatility

e¤ect yields the results in Proposition (2b:i) :

The result in Proposition (2b:ii) also follows from how the posterior mean versus vari-

ance e¤ect trade-o¤ is in�uenced by prior beliefs about default probability. With a high

prior default probability, the positive relation between posterior equity price, . P1E (z), and

posterior default probability occurs only for moderate values of public news, a region over

which the posterior variance e¤ect is muted.

A major implication of results in part (b) of Proposition (2) is that even though markets

update their posterior beliefs based on the public signal Y at t = 1; the relation between

security prices and posterior default probability continue to be signi�cantly in�uenced by

prior beliefs (at t = 0) about default likelihood. This again is a distinct feature of our setting

because the markets use the public signal Y to update their priors about both the mean and

the variance. In contrast, note that in the setting of part (a) of Proposition (2) when the

markets use the public signal to update only their prior beliefs about the mean, the prior

beliefs cease to play any role in determining the security prices�relation to posterior default

probability once the markets make their Bayesian updating.

13

Figure 1 below illustrates that, as one would expect, when the cash �ow X and the public

signal Y follow a joint normal distribution (as in Fischer and Verrecchia, 1997), the posterior

default probability monotonically decreases in the positively correlated public signal Z and

the posterior equity price following the public signal decreases monotonically in such signal

regardless of the prior default probability value. A similar behavior occurs with joint t

distribution only in the knife-edge case of prior default probability is exactly equal to 0:5:

This �gure assumes a speci�c form of the public signal Y = X + �:

Figure 1: Monotonic behavior of

posterior default probability and

equity price, P1E (z)

Assume Y = X + � and q = 0:8; �x = 0; � = 1:5; �x = 0:5; and �e = 0:5

The following �gures 2(a) and 2(b) revert to the more general form of a noisy, but posi-

tively correlated public signal as represented in Assumption 2 above. Figure 2(a) illustrates

Proposition (2b:i) result which deals with the setting of prior default probability being less

than 0:5 (i.e., when � < �x). In this case, �gure 2(a) illustrates the equity price P1E (z)

decreases along with posterior default probability for relative bad public news z and also

increases along with posterior default probability in the public signal z beyond the threshold

z�; the point at which the posterior default probability attains its minimum. The poste-

rior variance increasing signi�cantly in su¢ ciently positive public news z contributes to the

increase the posterior default, while at the same time the call option-like feature of the eq-

uity price increases in the volatility particularly given that the low prior default probability

mitigates the rate of increase in the posterior default probability in z: Similar intuition can

be adapted to the case of su¢ ciently low values of z as well because the posterior variance

14

signi�cantly increases for su¢ ciently low values of z as well.

Figure 2(a): Prop. 2(bi) result when prior

default probability is less than 0.5

Figure 2(b): Prop. 2(bii) result - when

prior default probability is greater than 0:5

Figure 2 (a) : k = 4; q = 0:8; �x = 2; � = 1:5; �x = 0:5 and �" = 1:8

Figure 2 (b) : k = 4; q = 0:8; �x = 0; � = 0:1; �x = 0:5 and �" = 0:5

In contrast, Figure 2(b) illustrates Proposition (2b:ii) result when the prior default prob-

ability is greater than 0:5 (i.e., when � > �x). Here, one can see that both the posterior

default probability and the posterior equity price P1E (z) decrease in moderately negative

public news z: Given the high prior default probability, even a moderately favorable public

news z is su¢ cient to reduce the posterior default probability, whereas the posterior variance

for such moderate public news z is not su¢ cient to increase the value of call option feature

embedded in equity prices to compensate for the decreasing �mean e¤ect� implied by the

bad news z. In this way, �gures 2(a) and 2(b) together illustrate the seemingly puzzling

result that equity prices can co-move in the same direction along with the posterior default

probability over di¤erent regions of the realized public news z; depending on the prior beliefs

about the default probability.

The next Proposition generates predictions about the �rm�s discretionary disclosure pro-

clivities as a function of some signi�cant parameters of our setting.

Proposition 3 The posterior probability of voluntary disclosure is:(a) increasing in the prior mean of the cash �ow �x;

(b) decreasing (increasing) in the prior volatility of the cash �ow, �X , if the prior default

15

probability is less (greater) than 0:5; and

(c) increasing (decreasing) in the correlation between the public signal and the liquidation

value, �XY , for relatively high (low) realized public news z.

The intuition for these comparative statics results follows. The result in Proposition (3:a)

is obvious. As the prior mean �x of the cash �ow increases, the posterior mean value also

increases and greater voluntary disclosure incentives to con�rm such posterior beliefs arise

naturally.

The intuition behind Proposition (3:b) is more complex. Observe that the default prob-

ability is� � �X � �XY �Xz

� (z)/ � � �X � �XY �Xz

�X=� � �X�X

� �XY z:

The above expression implies that the prior cash �ow variance has an e¤ect on voluntary

disclosure only through the change in the prior default probability on account of the public

news.

First, it can be established that in the knife-edge case of prior default probability be-

ing exactly equal to 0:5; the posterior probability of voluntary disclosure is independent of

the cash �ow volatility �x: In this case, the prior distribution is symmetric around zero;

so, though increasing variance thickens the tails, the distribution symmetry remains un-

changed yielding the result that voluntary disclosure probability being independent in cash

�ow volatility �x: When � � �x > 0; i.e., when prior default probability is greater than 0.5,then increasing the cash �ow volatility is good news because of a higher likelihood of cash

�ows being greater than the face value of debt �: So, a lower posterior default probability

increases the likelihood of voluntary disclosures.

Finally, the intuition behind Proposition (3:b) is straightforward. As the correlation

between the cash �owX and the public signal, Y; �XY increases, a good public news provides

greater incentives for the �rm to con�rm the public signal with his own credible voluntary

disclosures and vice versa.

While Proposition (3:) establishes monotonic voluntary disclosure behavior in prior pa-

rameters such as mean, variance and correlation coe¢ cient, the next Proposition predicts

that the public signal at t = 1 can induce non-monotonic voluntary disclosure behavior by

the �rm as a function of prior default probability.

Proposition 4 (a) If the prior default probability is less than 0:5, then there exists a thresh-old z� for the public news Z such that:

(ai) the posterior default probability is decreasing and probability of voluntary

disclosure is increasing in the public news z < z�; and

16

(aii) the posterior default probability is increasing and probability of voluntary

disclosure is decreasing in the public news z > z�.

(b) If the prior default probability is greater than 0:5, then there exists a threshold z�� for

the public news Z such that:

(bi) the posterior default probability is increasing and posterior probability of

voluntary disclosure is decreasing in the public news z < z��; and

(bii) the posterior default probability is decreasing and posterior probability of

voluntary disclosure is increasing in the public news z > z��.

Proposition (4) highlights the result that the e¤ect of public news on �rms�subsequent

voluntary disclosure behavior depends signi�cantly on prior beliefs about default probability.

When the prior default probability is less than 0:5 (i.e., when � < �x), Proposition

(4) shows that the posterior default probability is actually increasing in the public news

Z for su¢ ciently good news, i.e., for z > z�: When the prior default probability is low, a

su¢ ciently favorable surprise in public signal improves the posterior perception about the

cash �ow X signi�cantly, thus diminishing the incentives of the �rm to make a voluntary

disclosure for moderate values. Similarly, with a low prior default probability belief, the

�rm has incentives to make voluntary disclosures for low public news to improve the �rm�s

market price following voluntary disclosures at t = 2:

These voluntary disclosure incentives �ip when the prior default probability is high. The

�rm needs to con�rm good public news with its own voluntary disclosures, thus increasing the

likelihood of voluntary disclosures for su¢ ciently favorable public news. In this way, the prior

default probability actually serves as a de facto switch that turns the voluntary disclosure

as an informational �compliment�to the public news when prior default probability is high

and a �substitute�for the public signal with low prior default probability. This signi�cant

role of prior beliefs is driven by the market updating both the mean and variance based on

public news.

Figure 3 below illustrates that the cash �ow X and public signal Y have joint normal

distribution the posterior disclosure probability is monotonically increasing, while posterior

default probability is monotonically decreasing in the public news z regardless of the prior

default probability value. A similar behavior occurs with joint t distribution only in the

knife-edge case of prior default probability is exactly equal to 0:5:

17

Figure 3: Monotonic behavior of posterior

default and voluntary disclosure probabilities

Assume Y = X + �, q = 0:8; �x = 2; � = 1:5; �x = 0:5; and �e = 1:8

Figure 4 (a) below illustrates the setting in Proposition 4 (a) in which prior default probability

is less than 0:5 (i.e., when � < �x): Here, z� refers to the value of z at which the posterior

default probability attains its minimum. For values z < z�; the posterior default probability

decreasing and the posterior voluntary disclosure probability increasing in public news z

is intuitive as in �gure 3 above. However, for the public news z > z�, the posteriordisclosure probability is actually decreasing in z because of the greater posterior volatility

e¤ect increases the posterior default probability in a signi�cant enough manner to overcome

any expected bene�ts from disclosure.

18

Figure 4(a) : Proposition 4(a) result - when

prior default probability < 0:5

Figure 4(b) : Proposition 4(b) result - when

prior default probability > 0:5

Figure 4 (a) : k = 4; q = 0:8; �x = 2; � = 1:5; �x = 0:5 and �" = 1:8 yield z� = 1:07

Figure 4 (b) : k = 4; q = 0:8; �x = 0; � = 1:5:1; �x = 0:5 and �" = 1:8 yield z�� = �0:357

In contrast, �gure 4 (b) illustrates the setting in Proposition 4 (b) in which prior default

probability is greater than 0:5 (i.e., when � > �x): Here, z�� refers to the value of z at which

the posterior default probability attains its maximum. For values z < z��; the posterior

default probability actually is increasing and the posterior voluntary disclosure probability

decreasing in public news z because the �lingering e¤ects�of a high prior default probability

continue to dominate posterior updating e¤ects. However, for the public news z > z ��, theposterior default and disclosure probabilities behave in an intuitive manner. A comparison

�gures 4(a) and 4 (b) highlights our predictions about how posterior disclosure probability

varies in public news z as a function of prior default probabilities.

Corollary (1) next identi�es circumstances under which the posterior variance increases

everywhere in public news.

Corollary 1 The posterior probability of voluntary disclosure at t = 2 is monotonically

increasing and the posterior default probability is monotonically decreasing in the realized

value of the public news z when:

(a) the prior default probability is exactly equal to 0:5; given that the posterior variance

varies in the realized value of public signal Y (i.e., when the degrees of freedom k <1); or

19

(b) regardless of the prior default probability value, if the posterior variance does not vary

in the realized value of public signal Y (i.e., when the degrees of freedom k !1).

Part (a) of Corollary (1) deals with the knife-edge case of the prior default probability

being exactly equal to 0:5: In this case, the markets updating based on both the mean and

the variance would still lead to monotonically increasing disclosure likelihood because any

�residual memory�from prior beliefs post updating does not play any role in determining the

disclosure probability. Part (b) of Corollary (1) obtains a similar monotonically increasing

disclosure probability result because when k !1, we obtain the normal distribution whichdoes not allow for posterior variance to vary in the realized value of the public news Y .

Therefore, the information content of public news provides only a �compliment�role for the

�rm�s voluntary disclosures, with no scope for the �substitute�role, implying that the �rm�s

incentives to disclose increase everywhere in the public news Y .

The next Corollary seeks to provide one possible explanation for the gap between the

empirical literature (such as Miller, 2005) �ndings about better public news inducing greater

voluntary disclosures and prior theoretical work (such as ADK, and Einhorn , 2005)

Corollary 2 The probability of voluntary disclosure is higher conditional on relatively goodpublic news Y than conditional on relatively bad public news.

In prior disclosure models with unlimited liability (ADK and Einhorn 2005), public in-

formation does not a¤ect subsequent voluntary disclosure probability. But, Miller [2005]

empirically �nds that good public news actually prompts greater voluntary disclosures. A

combination of limited liability and posterior variance varying in the public news yields the

result in Corollary (2) :

5 Applications

All the results above predict how, in the presence of limited liability, the voluntary disclosure

strategy of a �rm is a¤ected by the realized public news and its statistical properties. The

above results do not hinge on a particular source of public information, which can be an

analyst report, a competitor�s announcement, or a mandatory disclosure of the �rm itself.

We next focus on two speci�c cases � the interaction between mandatory and voluntary

disclosures and analyst forecasts �and expand the set of empirical predictions generated by

our baseline model.

20

5.1 Mandatory and voluntary disclosures

Consider earnings announcements, when companies disclose their current earnings and, some-

times, contextually make a forecast about future earnings. By interpreting Y as a mandatory

disclosure, our baseline model can be immediately employed to study how the propensity to

make a voluntary disclosure depends on the characteristics of the mandatory environment.

Therefore, all prior �ndings can be rephrased in terms of mandatory disclosure of Y and

(either concurrent or following) voluntary disclosure of X.

Prior results (Proposition 2) primarily focused on properties of P1E (z) ; the interim

equity price at t = 1 following the surprise element in mandatory disclosure (hereafter,

simply referred to as mandatory disclosure) z; but before the �rm has an opportunity to

make a voluntary disclosure. In contrast, the following Proposition predicts the behavior of

P2E (fz; ndg) ; the equity price at time t = 2 following the mandatory disclosure of z and novoluntary disclosure (nd) by the �rm.

Proposition 5 Let z� and z�� denote the cuto¤s for the mandatory disclosures at t = 1; asidenti�ed by Propositions 4(a) and 4(b), respectively. The stock price following the realized

mandatory disclosure z and no voluntary disclosure, P2E (fz; ndg) ; is:(a) increasing in the prior mean of cash �ow, �X ;

(b) decreasing in the correlation between the cash �ow and the mandatory disclosure �XY ,

if the realized mandatory news z is relatively low;

(c) increasing in the mandatory disclosure z 2 (0; z�), if the prior default probability isless than 0:5;

(d) decreasing in the mandatory disclosure z < z�� and increasing in z > 0, if the prior

default probability is greater than 0:5;

(e) increasing in the prior volatility of the cash �ow �x if the prior default probability is

greater than or equal to 0:5; and

(f) increasing in mandatory disclosure z > 0 if the prior default probability is equal to

0:5.

The result in Proposition 5(a).is intuitive. As the prior mean �x of the cash �ows in-

creases, Proposition 5(a) states that for any given mandatory disclosure z; the market price

following no voluntary disclosure by the �rm, P2E (fz; ndg) ; would also increase. Recol-lect that the probability of the manager receiving private information is independent of the

realized value of cash �ow X, and hence, the market would reward the �rm with a better

non-disclosure price P2E (fz; ndg) as the prior mean �x increases. As the correlation betweenthe cash �ow and mandatory disclosure, �xy; increases, a low mandatory disclosure z would

only further con�rm the market�s posterior beliefs about the cash �ow x also being low;

21

hence, any non-disclosure by the �rm is viewed by the market more skeptically, leading to a

lower price P2E (fz; ndg) : This establishes the result in Proposition 5(b):Recollect from prior discussion that when the prior default probability is less than 0:5

(i.e., when � < �x), z� represents the value at which the posterior default probability reaches

its minimum. Proposition 5(c) states that the equity price following no voluntary disclosure,

P2E (fz; ndg) ; is increasing in z 2 (0; z�) because of a combination of two factors: (a)

conditioned on the manager not being privately informed, the value of equity claims are

increasing in z 2 (0; z�); and (b) the posterior default probability is also decreasing in

z 2 (0; z�) given that the prior default probability is less thank 0:5:Next, recollect that z�� represents the point at which the posterior default probability

attains its maximum value. When the prior probability is great than 0:5 (i..e., when � > �x),

Proposition 5(d) predicts that the equity price following no disclosure, P2E (fz; ndg) ; todecrease in z < z�� partly because of increasing default probability over that region.

Proposition 5(e) predicts that a higher cash �ow volatility �x would increase the non-

disclosure equity price P2E (fz; ndg) for any given z provided the prior default probability isgreater than or equal to 0:5: This result partly follows from the fact that conditioned on the

manager not being privately informed, the value of �option like�feature of equity of a levered

�rm increases in the cash �ow volatility �x:

Finally, Proposition 5(f) highlights the result that when the prior default probability is

exactly equal to 0:5; the monotonically increasing no-disclosure prices P2E (fz; ndg) re�ectthe setting with a normal distribution (i.e., when k !1) when the posterior variance doesnot vary in the mandatory disclosure z:

The above Proposition generated a variety of predictions about how a given level of equity

price following non-disclosure, P2E (fz; ndg) ; would change in a variety of parameters. Wenext proceed to generate additional predictions about how the stock price reaction following

non-disclosure. De�ne the scaled price reaction between date 1 and date 2 as

�(z; nd) � P2E (fz; ndg)� P1E (z)� (z)

: (5)

From our indi¤erence condition for disclosure threshold, it is clear that this price reaction

to the �rm�s non-disclosure is always negative for any given mandatory disclosure z because

non-disclosure induces the market�s skeptical beliefs. The next generation highlights non-

monotonic behavior of this stock price reaction following non-disclosure by the �rm following

the �rm�s mandatory disclosure z:

Proposition 6 The negative equity price reaction to the �rm�s non-disclosure relative to theprice following prior mandatory disclosure z appropriately scaled, �(z; nd) ; will be:

22

(a) U-shaped in the realized mandatory disclosure, z, if the prior default probability is

less than 0:5;

(b) hump-shaped in z, if the prior default probability is greater than 0:5;

(c) more negative if the prior mean, �X , increases;

(d) less negative (more negative) in the prior volatility, �X , if the prior default probability

is less (greater) than 0:5; and

(e) increasing (decreasing) in the correlation between the public signal and the liquidation

value, �XY , for relatively high (low) prior mandatory disclosure value z.

The intuition for the results in Proposition 6 follow closely to that of Proposition 4: It

can be shown that the stock price reaction to non-disclosure, �(z; nd) ; varies in mandatory

disclosure z at a rate that is proportional to the sensitivity of the normalized face value,~� = ���(z)

�(z)in z: Therefore, combining the feature that the stock price reaction to non-

disclosure, �(z; nd) ; is negative for all values of mandatory disclosure z; together with the

reasoning behind Proposition 4 results yields the above comparative statics predictions in

Proposition 6.

The above results are further clari�ed by the following set of illustrations. Figures 5 (a)

and 5 (b) depict that regardless of prior default probability value: (a) the stock price reaction

is non-monotonic in the quality of mandatory disclosures; and (b) further, as one would

expect, higher quality of mandatory disclosures cause the stock price reaction �(z; nd) to

non-disclosure more negative for less favorable mandatory disclosures z and more muted

(i.e., less negative) for favorable mandatory disclosures z than lower quality of mandatory

disclosures.

23

Figure 5(a) : E¤ect of mandatory

disclosure quality on stock price

reaction �(z; nd)

Figure 5(b) : E¤ect of mandatory

disclosure quality on stock price

reaction �(z; nd)

Figure 5 (b) : q = 0:k = 4; �x = 2; � = 1:5; and �x = 0:5

Figure 5 (b) : q = 0:k = 4; �x = 0; � = 1:5; and �x = 0:5

Figures 5 (c) and 5 (d) highlight the results from Proposition 6 (d) that a higher cash �ow

volatility has a muted (more negative) e¤ect on the negative stock price reaction to non-

disclosure, �(z; nd) ; when the prior default probability is less (greater) thank 0:5: With a

higher (lower) prior default probability, an increase in cash �ow volatility causes the market

to attribute the �rm�s non-disclosure to a greater (lower) chance of the �rm receiving less

favorable private information, which yield the relation illustrated in �gures 5 (c) and 5 (d) :

Further, these �gures illustrate that for any given cash �ow volatility, the negative stock

24

price reaction to non-disclosure, �(z; nd) ; is non-monotone.

Figure 5(c) : E¤ect of cash �ow

volatility on stock price reaction

�(z; nd)

Figure 5(d) : E¤ect of cash �ow

volatility on stock price reaction

�(z; nd)

Figure 5 (c) : q = 0:8; k = 4; �x = 2; � = 1:5; and �" = 1:8

Figure 5 (d) : q = 0:8; k = 4; �x = 0; � = 1:5; and �" = 1:8

Figure 5 (e) illustrates that the negative stock price reaction to non-disclosure, �(z; nd) ;

becomes stronger (i.e., more negative) when prior default probability is less than 0:5 than

when more than 0:5: Figure 5 (f) depicts a similar e¤ect with a higher prior mean than with

a lower one. In both cases, when the market has a more favorable prior belief (either in

terms of a lower default probability or a higher mean), the �rm�s non-disclosure imposes a

25

greater level of market skepticism, producing a starker negative response to non-disclosure.

Figure 5(e) : E¤ect of prior

default probability on stock

price reaction �(z; nd)

Figure 5(f) : E¤ect of cash �ow

mean on stock price reaction

�(z; nd)

Figure 5 (e) : q = 0:8; k = 4; � = 1:5; �x = 0:5; and �" = 1:8

Figure 5 (f) : q = 0:8; k = 4; � = 1:5; �x = 0:5; and �" = 1:8

The previous discussions in this section centered around equity price and price reaction

following no voluntary disclosure. We next proceed to examine whether the presence of a

mandatory disclosure requirement promotes or discourages voluntary disclosure probabil-

ity by comparing this regime with a mandatory disclosure requirement to the one with no

mandatory disclosure requirement. Figures 6 (a) and 6 (b) illustrate the role of mandatory

disclosure requirement in in�uencing subsequent voluntary disclosure probability. A com-

parison of �gures 6 (a) and 6 (b) reveals three interesting features. First, when prior default

probability is less than 0:5; �gure 6 (a) below illustrates that posterior voluntary disclosure

probability in the presence of mandatory disclosures is mostly lower than such discretionary

disclosure probability in the absence of such mandatory requirement, except for a small

neighborhood around the maximum point of posterior voluntary disclosure probability in

the presence of the mandatory disclosure requirement.

26

Figure 6 (a) : When prior default probability

< 0:5

Figure 6 (b) : When prior default probability

> 0:5

Figure 6 (a) : q = 0:8; k = 4; �x = 2; � = 1:5; �x = 0:5 and �" = 1:8

Figure 6 (b) : q = 0:8; k = 4; �x = 0; � = 1:5:1; �x = 0:5 and �" = 1:8

Second, when prior default probability is greater than 0:5; �gure 6 (b) illustrates that

posterior voluntary disclosure probability in the presence of a mandatory disclosure require-

ment is mostly greater than such discretionary disclosure probability in the absence of such

mandatory disclosures, except for a small neighborhood around the minimum point of poste-

rior voluntary disclosure probability in the presence of the mandatory disclosure requirement.

Finally, in the absence of a mandatory disclosure requirement, the voluntary disclosure prob-

ability is signi�cantly lower when the prior default probability is greater than 0:5 than when

the prior default probability is less than 0:5: Thus, a comparison of �gures 6 (a) and 6 (b)

suggests that the answer to the question whether a mandatory disclosure requirement en-

courages or discourages voluntary disclosures depends on whether prior default probability

is less than 0:5: If so, a mandatory disclosure requirement actually decreases the disclosure

likelihood for su¢ ciently large and small values of public news.

5.2 E¤ect of analyst forecast consensus and dispersion on volun-

tary disclosures and equity prices

Extend the previous model to the case where the public information is an n-dimensional

vector of analyst forecasts, for n � 2.9 Suppose that there are n symmetric analysts, indexedby i, each of whom receives a private signal of the form Si = X + Ei, where Ei is a zero-

9We require at least two analysts because with only one analyst the forecast dispersion is necessarily zero.

27

mean error term. We assume that (X;E1; : : : ; En) are uncorrelated and jointly distributed

according to a multivariate t distribution with k degrees of freedom. The variance of X

is �X and the variance of Ei is �E. After observing his private signal, analyst i issues an

unbiased forecast, Yi, of the �rm�s liquidation value X. Exploiting the properties of the

elliptical distributions, the forecast is given by

Yi = �X +�X

�X + �E(Si � �X) :

The vector of forecasts Y � (Y1; : : : ; Yn) is informationally equivalent to the vector of privatesignals S � (S1; : : : ; Sn). Therefore, to �nd the conditional distribution of XjS we can startfrom the joint distribution of (X; Y ). Letting �n be an n-dimensional vector of ones and Inbe the n-dimensional identity matrix, we have

X

Y

!� tn+1

�X

�X�n

!;

�X �XY

�0XY �Y

!; k

!;

where

�XY =�2X

�X + �E�0n;

�Y =

��X

�X + �E

�2(�EIn + �X�n�

0n) :

With this notation, the mean and variance of cash �ow X conditional on the set of analysts�

reports Y are given by

� (Y ) = �X +n (�X + �E)

�E + n�X(C � �X) ; (6)

�2 (Y ) =k

k + n

�E�X�E + n�X

+n

k + n

(�E + �X)2

�X (�E + n�X)

"D +

�E (C � �Y )2

�E + n�X

#; 10 (7)

where

C �Pn

i=1 Yin

and D �Pn

i=1 (Yi � C)2

n

are analyst forecast consensus (C) and forecast dispersion (D), respectively, following Barron

et al. (1998, 1999).

The next Proposition now predicts the �rm�s posterior voluntary disclosure probability

as a function of analysts�forecast consensus C and disperson D:

10For a derivation, see the proof of Proposition 7.

28

Proposition 7 The posterior probability of voluntary disclosure is:(a) U-shaped in consensus, C, if the prior default probability is greater than 0:5;

(b) hump-shaped in consensus, C; if the prior default probability is less than 0:5; and

(c) decreasing (increasing) in forecast dispersion, D, if the posterior default probability is

less (greater) than 0:5.

Propositions 7 (a) and 7 (b) results follow for similar reasons as in Propositions 4(a)

and 4(b) ; respectively. Further, Proposition 7 (c) result follows for similar reasons as in

Proposition 3(b) :

While the use of improper priors has several severe limitations, it is interesting to note

that the next Corollary generates predictions that are similar to the one with a conjugate

distribution whose posterior precision does not vary in the realized value of the public signal.

Corollary 3 When the prior on the liquidation value X is improper (i.e., as �X !1), theposterior probability of voluntary disclosure is:

(a) increasing everywhere in consensus C; and

(b) increasing (decreasing) in the precision of analysts�information, ��1E , if the posterior

default probability is less (greater) than 0:5.

Figures 7 (a) and 7 (b) below depict how posterior disclosure probability varies in analysts�

forecast consensus and dispersion, respectively.

29

Figure 7 (a) : E¤ect of

analysts�forecast consensus

on voluntary disclosure

probability

Figure 7 (b) : In�uence of

analysts�forecast dispersion on

voluntary disclosure probability

Figure 7 (a) : q = 0:8; k = 4; � = 1:5; �x = 0:5 and �" = 1:8

Figure 7 (b) : q = 0:8; n = 5; k = 3; �x = 8; � = 1:5:1; �x = 0:7 and �" = 4

It is interesting to note from �gure 7 (a) that for su¢ ciently low and high values of analyst

consensus values, posterior disclosure probability is greater when prior default probability

is less than 0:5 than when more than 0:5: Only for moderate values of consensus around

the prior mean of cash �ows we get a higher posterior disclosure probability with a prior

default probability being greater than 0:5: In other words, when prior beliefs about default

probability are low, it takes a su¢ ciently strong deviation of consensus number away from

the prior mean of cash �ow to induce the manager to disclose more often; otherwise, the

manager�s nondisclosure elicits relatively less market skepticism, thereby reducing their vol-

untary disclosure incentives. Figure 7 (b) illustrates two results: (i) �rst, that the posterior

disclosure probability is greater for every dispersion value when posterior default probability

is less than 0:5 than when greater than 0:5; and (ii) that the posterior disclosure probability

decreases (increases) in dispersion valueD when posterior default probability is less (greater)

than 0:5. The intuition for the �rst result (i) from �gure 7 (a) is straightforward and di-

rectly follows from applying Proposition 1 result that the posterior disclosure probability

is inversely related to the posterior default probability. The intuition for the second result

(ii) from �gure 7 (b) is a bit more involved and follows next. When the posterior default

probability is less than 0:5; the numerator in the posterior default probability G����x�(Y )

�is

negative and the denominator increases in the dispersion value D; thus leading to a lower

30

negative fraction. Consequently, the CDF G (�) increases in dispersion D; thus increasingthe posterior default probability, which feature in turn, leads to a lower posterior disclosure

probability from applying the result in Proposition 1: In contrast, when the posterior default

probability is greater than 0:5; the numerator in the posterior default probability G����x�(Y )

�is

positive. So, as the denominator increases in the dispersion value D; the CDF G (�) decreasesin dispersion D; thus decreasing the posterior default probability. Hence, from Proposition

1 it follows that the posterior disclosure probability increases in dispersion D with a high

enough posterior default probability. It must be noted that �gure 7 (b) illustrates the e¤ect of

a change in dispersion, holding a given consensus level C �xed. Further, it must be pointed

out that �gure 7 (b) is the only setting in which the comparative statics representation varies

in posterior (and not prior) default probability.

6 Conclusion

We are currently working on several extensions of this setting including examining the be-

havior of debt prices, design of debt contracts, managers�di¤erent objective functions, cost

of capital, dynamic disclosures and other related topics. In fact, a companion paper exam-

ines the �rm�s disclosure dynamics in the presence of possible loan sales by the lender in a

secondary market.

Appendix

Proof of Proposition 1. Using integrals, Equation (2) can be rewritten as

x̂� � =(1� q)

R1�(x� �) fXjY=y (x) dx+ q

R x̂�(x� �) fXjY=y (x) dx

(1� q) + qF (x̂) :

Next, note that P [X < xjY ] = G�x���

�, where G (�) is the cumulative function of U , and so

fXjY=y (x) =1�g�x���

�, where g (�) is the density function of U . By applying the change of

variable x = �+ �u to the right-hand side, we obtain

x̂� � =(1� q)

R1����(�+ �u� �) g (u) du+ q

R x̂���

����

(�+ �u� �) g (u) du

(1� q) + qG�x̂���

� :

31

Further rearranging, we have

~x� ~� =(1� q)

hR1�1

�u� ~�

�g (u) du�

R ~��1

�u� ~�

�g (u) du

i+ q

R ~x~�

�u� ~�

�g (u) du

(1� q) + qG (~x) ;

where

~x � x̂� ��

; ~� � � � ��

are the �standardized�disclosure threshold and face value of the debt. Using integration by

parts, the indi¤erence condition becomes

~x� ~� =(1� q)

h�~� +

R ~��1G (u) du

i+ q

h�~x� ~�

�G (~x)�

R ~x~�G (u) du

i(1� q) + qG (~x) ;

and thus

(1� q)"~x�

Z ~�

�1G (u) du

#+ q

Z ~x

~�

G (u) du = 0: (8)

The posterior disclosure probability is q [1�G (~x)], hence it is decreasing in the equilib-rium standardized disclosure threshold ~x. The posterior default probability is G

�~��, which

is increasing in the standardized face value ~�. We then show that the equilibrium ~x is an

increasing function of ~�. To this purpose, totally di¤erentiate (8) with respect to y,

(1� q)h~x0 �G

�~��~�0i+ q

hG (~x) ~x0 �G

�~��~�0i= 0;

whence

~x0 =G�~��

(1� q) + qG (~x)~�0: (9)

Since sign (~x0) = sign�~�0�, the posterior disclosure probability is higher when ~� is lower.

Proof of Remark 1. An increase in the amount of borrowing M is mathematically

equivalent to an increase in �. As � increases, ~� increases and, by Proposition 1, the disclosure

probability decreases.

Proof of Remark 2. Suppose that (X; Y ) follow a joint multivariate t distribution with

k degrees of freedom, and that Y and X are nY -dimensional and one-dimensional random

vectors, respectively. De�ne

� (y) � �X � �XY��1Y (y � �Y )

�2 (y) ���X � �XY��1Y �0XY

��k + (y � �Y )0��1Y (y � �Y )k + nY

�:

32

Then,

Xj (Y = y) � tnX�� (y) ; �2 (y) ; k + nY

�;

that is, the posterior distribution of X is also multivariate t (e.g., Ding (2016)).

If we let Uk+nY denote a standard univariate t distribution with k+nY degrees of freedom,

by the properties of a¢ ne transformation of elliptically distributed random variables (e.g.,

Gomez et al. (2003)),

� (y) + � (y)Uk+nY � tnX�� (y) ; �2 (y) ; k + nY

�;

and therefore Xj (Y = y) is equal in distribution to � (y) + � (y)Uk+nY .Proof of Proposition 2. We �rst derive the expression in (4). Let fXjZ=z (�) and FXjZ=z (�)denote the posterior density and cumulative functions for X. The posterior equity value is

E [VE (X) jZ = z] =Z 1

�

(x� �) fXjZ=z (x) dx =Z 1

�

(x� �) 1�gk

�x� ��

�dx

= �

Z 1

~�

�u� ~�

�gk (u) du;

where: the �rst equality follows from the de�nition; the second equality from invoking the

fact that FXjZ=z (x) = Gk�x���

�and taking the derivative of Gk

�x���

�with respect to x; the

third from operating the change of variable u = x���.

Proof of Part (a). Note that k ! 1, �0 (z) > 0 but �0 (z) = 0. We begin by showing

that the default probability is decreasing in the realized news z, namely, that ~�0(z) < 0.

This follow from ~�0= ��0

�< 0. Next, we show that E [VE (X) jZ = z] is increasing in z:

@E [VE (X) jZ = z]@z

= �0Z 1

~�

�u� ~�

�gk (u) du+ �

���~� � ~�

�gk (u)�

Z 1

~�

gk (u) du

�~�0

= ��h1�Gk

�~��i~�0> 0:

Proof of Part (b.i). From the proof of Proposition 4 we know that when the prior default

probability is lower than 0:5 we have ~�0(z) < 0 for z < z� and ~�

0(z) > z�. From the proof

of Proposition 2 we also know that limz!�1 ~� (z) = l > 0 and limz!1 ~� (z) = �l < 0, wherel <1. Taking the derivative of � (z) with respect to the realized news z gives

�0 (z) = �X

s1� �2XYk + 1

zpk + z2

;

and therefore: �0 (z) < 0 for z < 0 and �0 (z) > 0 for z > 0; limz!�1 �0 (z) = ��XY �X=l < 0

33

and limz!1 �0 (z) = �XY �X=l > 0. Also, consider the product

� (z) ~�0(z) =

24�Xs1� �2XYk + 1

pk + z2

352664��XY �X

qk+z2

k+1� (���X��XY �Xz) z

k+1qk+z2

k+1

�Xp(1� �2XY )k+z

2

k+1

3775=

��XY �Xk � (� � �X) z(k + z2)

;

hence limz!�1 � (z) ~�0(z) = limz!1 � (z) ~�

0(z) = 0.

The derivative of the posterior equity value is

@E [VE (X) jZ = z]@z

= �0Z 1

~�

�u� ~�

�gk (u) du� �~�

0 h1�Gk

�~��i;

and so, using the limit properties above,

limz!�1

@E [VE (X) jZ = z]@z

= ��XY �Xl

Z 1

l

(u� l) gk (u) du < 0

limz!1

@E [VE (X) jZ = z]@z

=�XY �Xl

Z 1

�l(u+ l) gk (u) du > 0:

Thus, in these limits the derivative of the posterior equity value has the same sign as ~�0(z).

Proof of Part (b.ii). From the proof of Proposition 4 we know that when the prior default

probability is greater than 0:5 we have ~�0(z) > 0 for z < z�� and ~�

0(z) < z�� for z > z��,

where the cuto¤ z�� < 0. Therefore, �0 (z��) < 0 and ~�0(z��) = 0 imply (by continuity)

that E [VE (X) jZ = z] is decreasing in a neighborhood to the right of z��, where the defaultprobability is also decreasing.

Proof of Part (b.iii). The claim follows from Lemma 1.

Proof of Proposition 3. To prove the results, we take the derivative of ~� with respect to

the various parameters.

Proof of Part (a). By inspection of ~�.

Proof of Part (b). We have ~�0(�X) / � (� � �X).

Proof of Part (c). Di¤erentiation with respect to �XY yields

~�0(�XY ) / ��Xz

q(1� �2XY ) + (� � �X � �XY �Xz)

�XYp(1� �2XY )

/ (� � �X) �XY � �Xz:

For z ! �1, ~�0 (�XY ) is positive, whereas for z !1, ~�0 (�XY ) is negative.

34

Proof of Proposition 4. The prior default probability is P [X < �] = Gk

����Xp�X

�, where

Gk (�) is the cumulative function of the standard univariate t with k degrees of freedom.This claim follows from the fact that the marginal on X of the joint distribution of (X;Y )

is the univariate t1 (�X ;�X ; k). Since the t distribution is symmetric around its mean,

P [X < �] > 0:5 if and only if � � �X > 0.Proof of Part (a). Take the derivative of ~� with respect to z,

~�0(z) =

��XY �Xq

k+z2

k+1� (���X��XY �Xz) z

k+1qk+z2

k+1

�Xp(1� �2XY )k+z

2

k+1

;

so ~�0(z) > 0 if and only if (� � �X) z < ��XY �Xk. When � � �X < 0, ~�

0(z) > 0 if and only

if z > z� � �XY �Xk�X��

. Therefore, for realizations z < z� (z > z�) the probability of disclosure

is increasing (decreasing) in z, and maximized at z = z�.

Proof of Part (b). When ���X > 0, ~�0(z) > 0 if and only if z < z�� � ��XY �Xk

���X. Hence,

for realizations z < z�� (z > z��) the probability of disclosure is decreasing (increasing) in z,

and minimized at z = z��.

Proof of Corollary 1. When � � �X = 0, ~�0(z) / ��XY �Xk < 0, and so the disclosure

probability is always increasing in z.

Proof of Corollary 2. By Proposition 1, the posterior disclosure probability is a decreas-

ing function of the posterior default probability. Thus, we study the limits of the default

probability as z ! �1 and z ! 1, and show that the default probability is lower in thelatter limit.

Given Assumption 2 on a joint t distribution, the posterior default probability is

P [X < �jZ = z] = P [� (Z) + � (Z)Uk+1 < �jZ = z] = Gk+1�~��;

where~� � � � �X � �XY �Xz

�X

q(1� �2XY ) k+z

2

k+1

;

and Gk+1 (�) is the cumulative function of a standard univariate t distribution with k + 1degrees of freedom. The claim follows because

limz!�1

~� = l > limz!1

~� = �l;

where l � �XY =q(1� �2XY ) (k + 1)

�1 > 0.

35

Proof of Proposition 5. First, note that P (fnd; zg) = ��~x� ~�

�. Hence, taking the

derivative with respect to the news z or any parameter,

P (fnd; zg)0 = �0�~x� ~�

�+ �

�~x0 � ~�0

�= �0

�~x� ~�

�� ~�0�

241� G�~��

(1� q) + qG (~x)

35= �0

24~x� G�~��

(1� q) + qG (~x)~�

35+241� G

�~��

(1� q) + qG (~x)

35�0;where the second equality follows from (9) and the last equality from ~�

0= 1

�

���0 � �0~�

�.

Proof of Part (a). �0 (�X) = 0 and �~�0(�X) > 0 (by Proposition 3(a)).

Proof of Part (b). �0 (�XY ) < 0 and �~�0(�XY ) < 0 for z su¢ ciently low (by Proposition

3(d)).

Proof of Part (c). Note that, in equilibrium, it holds that ~x > ~�. This inequality implies

G�~��< (1� q) + qG (~x). Thus, when �0 and ~�0 have opposite signs, the sign of P (fnd; zg)0

is equal to the sign of �0. The posterior volatility is U-shaped z and achieves its minimum

at z = 0. When the prior default probability is less than 0:5, ~�0< 0 to the left of z� (by

Proposition 4(a)). Since z� > 0, for z 2 (0; z�) sign (�0) = �sign�~�0�> 0, which, by our

previous consideration, implies P (fnd; zg)0 = sign (�0) > 0 in this interval.Proof of Part (d). When the prior default probability is greater than 0:5, ~�

0> 0 to the

right of z�� (by Proposition 4(b)). In particular, ~�0> 0 for z > 0, since z�� < 0. For z < z��,

sign (�0) = �sign�~�0�< 0, and so P (fnd; zg)0 = sign (�0) < 0. Vice versa, for z > 0,

sign (�0) = �sign�~�0�> 0, and so P (fnd; zg)0 = sign (�0) > 0.

Proof of Part (e). �0 (�X) > 0 and �~�0 (�X) > 0 when the prior default probability is

greater than 0:5 (by Proposition 3(b)).

Further, �0 (�X) > 0 and �~�0 (�X) = 0 when the prior default probability is 0:5 (by

Proposition 3(c)).

Proof of Part (f). When the prior default probability is equal to 0:5, ~�0< 0 for all z

(by Proposition 4(c)). Therefore, for z > 0 we have sign (�0) = �sign�~�0�> 0, and so

P (fnd; zg)0 = sign (�0) > 0.Proof of Proposition 6. Note that P (fnd; zg) = �

�~x� ~�

�and P (z) = �

R1~�

�u� ~�

�gk (u) du.

36

Using integration by parts, the posterior equity price can be written as

P (z) = �

"Z 1

�1

�u� ~�

�gk (u) du�

Z ~�

�1

�u� ~�

�gk (u) du

#

= �

"�~� +

Z ~�

�1Gk (u) du

#:

Therefore,P2E (fz; ndg)� P1E (z)

� (z)= ~x�

Z ~�

�1Gk (u) du:

Its derivative is

~x0 �Gk�~��~�0= ~�

0Gk

�~��� 1

(1� q) + qGk (~x)� 1�:

Because (1� q) + qGk (~x), the sign of the scaled price reaction is the same as the sign of ~�0.

The rest of the proof follows along prior lines.

Proof of Proposition 7. We �rst derive Equations (6) and (7). Let �n be the projection

matrix �n � �n (�0n�n)�1 �0n. Then,

�Y =

��X

�X + �E

�2[�E (�n + In � �n) + n�X�n]

=

��X

�X + �E

�2[(�E + n�X)�n + �E (In � �n)] ;

hence

��1Y =

��X

�X + �E

��2 �1

�E + n�X�n +

1

�E(In � �n)

�:11

To compute the posterior mean, observe that

� (Y ) = �X + �XY��1Y (Y � �X�n)

= �X + (�X + �E) �0n

�1

�E + n�X�n +

1

�E(In � �n)

�(Y � �X�n)

= �X + (�X + �E)�0n (Y � �X�n)�E + n�X

= �X + (�X + �E)n (C � �X)�E + n�X

;

11One sees that this expression corresponds to the inverse of �Y by checking that �Y ��1Y = In. The latter

identity is due to the following properties of the projection matrix: �n is idempotent (hence �n�n = �n);and �0n (In ��n) = 0 implies �n (In ��n) = 0.

37

which has exploited the fact that �n (In � �n) = 0. As to the posterior variance, note that

�X � �XY��1Y �0XY =�E�X

�E + n�X;

and

(Y � �Y �)0�

1

�E + n�X�n +

1

�E(In � �n)

�(Y � �Y �)

=

�Y 0�

1

�E + n�X�n +

1

�E(In � �n)

�� �Y

1

�E + n�X�0�(Y � �Y �)

=1

�E + n�XY 0�nY +

1

�EY 0 (In � �n)Y � 2

�Y�E + n�X

Y 0�+n�2Y

�E + n�X

=nC2

�E + n�X+nD

�E� 2 �Y nC

�E + n�X+

n�2Y�E + n�X

= n

"D

�E+(C � �Y )

2

�E + n�X

#:

Hence,

(Y � �Y �)0��1Y (Y � �Y �) =

��E + �X�X

�2n

"D

�E+(C � �Y )

2

�E + n�X

#and

�2 (Y ) =�E�X

�E + n�X

k +��E+�X�X

�2nhD�E+ (C��Y )2

�E+n�X

ik + n

=

�E�X�E+n�X

k + n(�E+�X)2

�X(�E+n�X)

hD + �E(C��Y )2

�E+n�X

ik + n

:

lim�X!1

��1Y =1

�E(In � �n) ;

we have

lim�X!1

(Y � �Y �n)0��1Y (Y � �Y �n) = ��1E (Y � �Y �n)

0 (In � �n) (Y � �Y �n)

= ��1E [Y 0 (In � �n)� �Y �0n (In � �n)] (Y � �Y �n)= ��1E Y

0 (In � �n)Y = ��1E nD:

Proof of Parts (a and b). By inspection of ~� we see that the default probability decreases

38

if and only if �0 (C)� + �0 (C) (� � �) > 0. Observe that

�0 (C) =n (�X + �E)

�E + n�X;

�0 (C) =

nk+n

�E(�E+�X)2

�X(�E+n�X)2 (C � �Y )

�:

It follows that ~�0(C) < 0 if and only if

0 <n (�X + �E)

�E + n�X

(k

k + n

�E�X�E + n�X

+n

k + n

(�E + �X)2

�X (�E + n�X)

"D +

�E (C � �Y )2

�E + n�X

#)

+n

k + n

�E (�E + �X)2

�X (�E + n�X)2 (C � �Y )

�� � �X �

n (�X + �E)

�E + n�X(C � �X)

�;

that is, if and only if

0 <

�k

�2X�E + �X

+ n�E + �X�E

D

�+ (� � �X) (C � �Y ) : (10)

The claims follows from inspection of (10).

Proof of Part (c). The posterior default probability is less than 0:5 if and only if its

numerator, � � � (Y ), is negative. In that case, increasing D at the denominator leads to

an increase in ~� (namely, ~� becomes less negative). Therefore, if � � � (Y ) < 0 the defaultprobability is higher when the dispersion is higher. As a consequence, the disclosure proba-

bility is lower when � � � (Y ) < 0 and D is greater. Oppositely, the disclosure probability

is greater when � � � (Y ) > 0 and D is lower, since this combination corresponds to a lower

probability of default. When � � � (Y ) = 0, changes in the denominator do not a¤ect theposterior default probability.

Proof of Corollary 3. When �X ! 1, the mean and variance of the liquidation valueconditional on the analysts�reports are given by

� (Y ) = C and �2 (Y ) =�E

kn+D

k + n: (11)

The claims follow from inspection of ~� evaluated using the expressions in (11).

39

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